2

An electromagnetic wave (like a propagating photon) is known to carry it's electric and magnetic field-vectors perpendicular and each depending on the differential change of the other thus "creating" each other and therefore appearing in-phase and reaching their minima/maxima together. I'm interested to know whether there was any uncertainty as a principle discussed in the underlying principles of the Maxwell equations, like $\Delta E \Delta B \geq \hbar$. I appreciate links, hints and answers.

Waffle's Crazy Peanut
  • 9,088
JuSchu
  • 21
  • 1
    Classical, i.e. Maxwellian, Electrodynamics is a non-quantized theory, so it's not bound to have this kind a heisenberg-like uncertainty relation. QED have this kind of relation for Electromagnetic field amplitudes.

    Remember that you can have a electrostatic field configuration(think on a uniformely charged) sphere that have (classically) $\vec B=0$ in the whole space, and also you have a well defined $\vec E$ so both $\Delta E$ and $\Delta B$ are zero.

    Also remember that the electromagnetic field is a single object, in both classical and quantum electromagnetic theory.

     – Hydro Guy May 07 '13 at 15:03
  • @user23873 All right, I have to admit that classical mechanics always bothered me because of it's idiosyncratic assumptions. I think I'll have to have a deep look in a Feynman lecture of QED. – JuSchu May 07 '13 at 15:53

2 Answers2

3

One can see the consistency with the Heisenberg Uncertainty Principle by the definition of wavelength and frequency of the electromagnetic wave:

$\lambda\nu/c=1$ where c is the velocity of light

Multiplying both sides by $h$ and considering $\lambda$ as $\delta(x)$ and $p=h\nu/c$ for a photon,

$$\lambda h\nu/c \sim h$$

$$\delta(x)\delta(p)\sim h$$

Dev
  • 317
anna v
  • 233,453
  • Yes, @anna v, but still, within this uncertainty of the electromagnetic wave in itself (as a hole), I can not see whether it could be theoretically possible to detect the magnitude of the electric and the magnetic vector simultaneously. – JuSchu May 08 '13 at 10:19
  • 1
    the photon is an elementary particle, it has no E or B in the sense of classical EM wave. If you want to see how the classical wave is built up by an ensemble of photons see this blog entry http://motls.blogspot.com/2011/11/how-classical-fields-particles-emerge.html – anna v May 08 '13 at 11:54
  • Thx @anna v. I was just trying to glue this with the statement "The electric and magnetic fields of a single photon in a box are in fact very important and interesting", being part of a broadly accepted answer to this question concerning the work of Serge Haroche. I am willing and ready to read and to learn and the road to reality needs good shoes. – JuSchu May 08 '13 at 15:30
  • @mods. A wrong mini-Markdown formatting occured. My fault. Did I miss a preview function on comments before adding my wrongly formatted link to the comment? How can I fix it? – JuSchu May 09 '13 at 15:17
  • @dmckee there is a comment for mods here – anna v May 09 '13 at 19:16
0

According to this Letter to the Editor from 1933, written by G. Lemaitre at MIT (or at that time University of Louvain) https://journals.aps.org/pr/abstract/10.1103/PhysRev.43.148 https://doi-org.ezproxy.library.ubc.ca/10.1103/PhysRev.43.148

$\Delta E_x\Delta H_z = hc/l^4$

"refers to the mean measures of two perpendicular components $E_x$, $H_z$ of the electric and magnetic fields in a cube of side $l$. It must be understood as referring to the time-mean values of this field during the time $l/c$."

It has only 2 citing articles, and likely became out-dated shortly after its publish date. For instance, one article which cites it in 2002,

https://doi-org.ezproxy.library.ubc.ca/10.1016/S1355-2198(02)00033-3

Claims: "But it is, in fact, not obvious that Lemaıtre in 1934 was unaware of the vacuum energy arising in quantum field theory. For instance he discusses Heisenberg uncertainty relations for the electromagnetic field in a short article from 1933 (Lemaıtre, 1933) in connection with the then newly formulated quantum principles for the electromagnetic field."

Since electric and magnetic fields are not conserved quantities like energy and momentum, they are not typically used in Heisenberg's uncertainty principle. More often position and momentum or energy and time are used as Anna V. suggests. However, variance relations exist for the ground-states of displacement fields $D$ and $B$ in a dielectric (1991): https://doi.org/10.1103/PhysRevA.43.467

$\Delta D\Delta B = \frac{1}{2}\hbar\omega$

The right-hand-side of the above equation is the "zero-point-energy" or "vacuum energy" for photons mentioned above. It occurs at a temperature of 0K because of Heisenberg's uncertainty relation.

Cael Warner
  • 1